<?xml version="1.0" encoding="UTF-8"?>
<sect1 id="reference_sfcgal">
	<sect1info>
		<abstract>
				<para>
          SFCGAL is a C++ wrapper library around CGAL that provides advanced 2D and 3D spatial functions.
          For robustness, geometry coordinates have an exact rational number representation.
        </para>
        <para>
          Installation instructions for the library can be found on the SFCGAL home page (<ulink url="http://www.sfcgal.org">http://www.sfcgal.org</ulink>).
          To enable the functions use <code>create extension postgis_sfcgal</code>.
        </para>

		</abstract>
	</sect1info>
	<title>SFCGAL Functions</title>
	<refentry id="postgis_sfcgal_version">
	  <refnamediv>
		<refname>postgis_sfcgal_version</refname>

		<refpurpose>Returns the version of SFCGAL in use</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>text <function>postgis_sfcgal_version</function></funcdef>
                        <void/>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

        <para>Returns the version of SFCGAL in use</para>

		<para role="availability" conformance="2.1.0">Availability: 2.1.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>
	  <refsection>
				<title>See Also</title>

				<para><xref linkend="postgis_sfcgal_full_version" /></para>
		</refsection>

	</refentry>

	<refentry id="postgis_sfcgal_full_version">
	  <refnamediv>
		<refname>postgis_sfcgal_full_version</refname>

		<refpurpose>Returns the full version of SFCGAL in use including CGAL and Boost versions</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>text <function>postgis_sfcgal_full_version</function></funcdef>
                        <void/>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

        <para>Returns the full version of SFCGAL in use including CGAL and Boost versions</para>

		<para role="availability" conformance="3.3.0">Availability: 3.3.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>
	  <refsection>
				<title>See Also</title>

				<para><xref linkend="postgis_sfcgal_version" /></para>
		</refsection>

	</refentry>

    <refentry id="ST_3DArea">
	  <refnamediv>
		<refname>ST_3DArea</refname>

		<refpurpose>Computes area of 3D surface geometries.  Will return 0 for solids.</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>float<function>ST_3DArea</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom1</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

		<para role="availability" conformance="2.1.0">Availability: 2.1.0</para>
		<para>&sfcgal_required;</para>
        <para>&sqlmm_compliant; SQL-MM IEC 13249-3: 8.1, 10.5</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>

	  <refsection><title>Examples</title>
	  <para>Note: By default a PolyhedralSurface built from WKT is a surface geometry, not solid.  It therefore has surface area.  Once converted to a solid, no area.</para>
<programlisting>SELECT ST_3DArea(geom) As cube_surface_area,
	ST_3DArea(ST_MakeSolid(geom)) As solid_surface_area
  FROM (SELECT 'POLYHEDRALSURFACE( ((0 0 0, 0 0 1, 0 1 1, 0 1 0, 0 0 0)),
    ((0 0 0, 0 1 0, 1 1 0, 1 0 0, 0 0 0)),
    ((0 0 0, 1 0 0, 1 0 1, 0 0 1, 0 0 0)),
    ((1 1 0, 1 1 1, 1 0 1, 1 0 0, 1 1 0)),
    ((0 1 0, 0 1 1, 1 1 1, 1 1 0, 0 1 0)),
    ((0 0 1, 1 0 1, 1 1 1, 0 1 1, 0 0 1)) )'::geometry) As f(geom);

 cube_surface_area | solid_surface_area
-------------------+--------------------
                 6 |                  0    </programlisting>

	  </refsection>
	  <refsection>
		<title>See Also</title>

		<para><xref linkend="ST_Area" />, <xref linkend="ST_MakeSolid" />, <xref linkend="ST_IsSolid" />, <xref linkend="ST_Area" /></para>
	  </refsection>
	</refentry>

	<refentry id="ST_3DConvexHull">
	  <refnamediv>
		<refname>ST_3DConvexHull</refname>

		<refpurpose>Computes the 3D convex hull of a geometry.</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_3DConvexHull</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom1</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

		<para role="availability" conformance="3.3.0">Availability: 3.3.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>

	  <refsection>
	      <title>Examples</title>
         <programlisting>SELECT ST_AsText(ST_3DConvexHull('LINESTRING Z(0 0 5, 1 5 3, 5 7 6, 9 5 3 , 5 7 5, 6 3 5)'::geometry));</programlisting>
          <screen>POLYHEDRALSURFACE Z (((1 5 3,9 5 3,0 0 5,1 5 3)),((1 5 3,0 0 5,5 7 6,1 5 3)),((5 7 6,5 7 5,1 5 3,5 7 6)),((0 0 5,6 3 5,5 7 6,0 0 5)),((6 3 5,9 5 3,5 7 6,6 3 5)),((0 0 5,9 5 3,6 3 5,0 0 5)),((9 5 3,5 7 5,5 7 6,9 5 3)),((1 5 3,5 7 5,9 5 3,1 5 3)))</screen>

          <programlisting>WITH f AS (SELECT i, ST_Extrude(geom, 0,0, i ) AS geom
FROM ST_Subdivide(ST_Letters('CH'),5) WITH ORDINALITY AS sd(geom,i)
      )
      SELECT ST_3DConvexHull(ST_Collect(f.geom) )
      FROM f;</programlisting>
         <para>
            <informalfigure>
                <mediaobject>
                    <imageobject>
                        <imagedata fileref="images/st_3dconvexhull.png" />
                    </imageobject>
                    <caption><para>Original geometry overlaid with 3D convex hull</para></caption>
                </mediaobject>
            </informalfigure>
        </para>
	  </refsection>
      <refsection>
		<title>See Also</title>
		<para><xref linkend="ST_Letters" />, <xref linkend="ST_AsX3D" /></para>
	  </refsection>
	</refentry>

	<refentry id="ST_3DIntersection">
	  <refnamediv>
		<refname>ST_3DIntersection</refname>

		<refpurpose>Perform 3D intersection</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_3DIntersection</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom1</parameter></paramdef>
			<paramdef><type>geometry</type> <parameter>geom2</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>
		<para>Return a geometry that is the shared portion between geom1 and geom2.</para>
		<para role="availability" conformance="2.1.0">Availability: 2.1.0</para>
		<para>&sfcgal_required;</para>
        <para>&sqlmm_compliant; SQL-MM IEC 13249-3: 5.1</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>

	  <refsection>
		<title>Examples</title>
		<para>3D images were generated using PostGIS <xref linkend="ST_AsX3D" /> and rendering in HTML using <ulink url="http://www.x3dom.org">X3Dom HTML Javascript rendering library</ulink>.</para>
            <informaltable>
            <tgroup cols="2">
            <tbody>
              <row>
                <entry><para>
                <programlisting>SELECT ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(100 90)'),
 50, 'quad_segs=2'),0,0,30) AS geom1,
        ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(80 80)'),
 50, 'quad_segs=1'),0,0,30) AS geom2;
                </programlisting>
                <informalfigure>
                    <mediaobject>
                      <imageobject>
                        <imagedata fileref="images/st_3ddifference01.png" />
                      </imageobject>
                      <caption><para>Original 3D geometries overlaid. geom2 is shown semi-transparent</para></caption>
                    </mediaobject>
                  </informalfigure></para>
              </entry>
              <entry><para>
              <programlisting>SELECT ST_3DIntersection(geom1,geom2)
FROM ( SELECT ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(100 90)'),
 50, 'quad_segs=2'),0,0,30) AS geom1,
        ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(80 80)'),
 50, 'quad_segs=1'),0,0,30) AS geom2 ) As t;</programlisting>
              <informalfigure>
                    <mediaobject>
                      <imageobject>
                        <imagedata fileref="images/st_3dintersection01.png" />
                      </imageobject>
                      <caption><para>Intersection of geom1 and geom2</para></caption>
                    </mediaobject>
                  </informalfigure></para>
              </entry>
              </row>
             </tbody>
            </tgroup>
        </informaltable>

		<para>3D linestrings and polygons</para>
		<programlisting>	SELECT ST_AsText(ST_3DIntersection(linestring, polygon)) As wkt
FROM  ST_GeomFromText('LINESTRING Z (2 2 6,1.5 1.5 7,1 1 8,0.5 0.5 8,0 0 10)') AS linestring
 CROSS JOIN ST_GeomFromText('POLYGON((0 0 8, 0 1 8, 1 1 8, 1 0 8, 0 0 8))') AS polygon;

              wkt
--------------------------------
 LINESTRING Z (1 1 8,0.5 0.5 8)
		</programlisting>

		<para>Cube (closed Polyhedral Surface) and Polygon Z</para>
		<programlisting>SELECT ST_AsText(ST_3DIntersection(
		ST_GeomFromText('POLYHEDRALSURFACE Z( ((0 0 0, 0 0 1, 0 1 1, 0 1 0, 0 0 0)),
	((0 0 0, 0 1 0, 1 1 0, 1 0 0, 0 0 0)), ((0 0 0, 1 0 0, 1 0 1, 0 0 1, 0 0 0)),
	((1 1 0, 1 1 1, 1 0 1, 1 0 0, 1 1 0)),
	((0 1 0, 0 1 1, 1 1 1, 1 1 0, 0 1 0)), ((0 0 1, 1 0 1, 1 1 1, 0 1 1, 0 0 1)) )'),
	'POLYGON Z ((0 0 0, 0 0 0.5, 0 0.5 0.5, 0 0.5 0, 0 0 0))'::geometry))</programlisting>
		<screen>TIN Z (((0 0 0,0 0 0.5,0 0.5 0.5,0 0 0)),((0 0.5 0,0 0 0,0 0.5 0.5,0 0.5 0)))</screen>

		<para>Intersection of 2 solids that result in volumetric intersection is also a solid (ST_Dimension returns 3)</para>
		<programlisting>SELECT ST_AsText(ST_3DIntersection( ST_Extrude(ST_Buffer('POINT(10 20)'::geometry,10,1),0,0,30),
 ST_Extrude(ST_Buffer('POINT(10 20)'::geometry,10,1),2,0,10) ));</programlisting>
	<screen>POLYHEDRALSURFACE Z (((13.3333333333333 13.3333333333333 10,20 20 0,20 20 10,13.3333333333333 13.3333333333333 10)),
	((20 20 10,16.6666666666667 23.3333333333333 10,13.3333333333333 13.3333333333333 10,20 20 10)),
	((20 20 0,16.6666666666667 23.3333333333333 10,20 20 10,20 20 0)),
	((13.3333333333333 13.3333333333333 10,10 10 0,20 20 0,13.3333333333333 13.3333333333333 10)),
	((16.6666666666667 23.3333333333333 10,12 28 10,13.3333333333333 13.3333333333333 10,16.6666666666667 23.3333333333333 10)),
	((20 20 0,9.99999999999995 30 0,16.6666666666667 23.3333333333333 10,20 20 0)),
	((10 10 0,9.99999999999995 30 0,20 20 0,10 10 0)),((13.3333333333333 13.3333333333333 10,12 12 10,10 10 0,13.3333333333333 13.3333333333333 10)),
	((12 28 10,12 12 10,13.3333333333333 13.3333333333333 10,12 28 10)),
	((16.6666666666667 23.3333333333333 10,9.99999999999995 30 0,12 28 10,16.6666666666667 23.3333333333333 10)),
	((10 10 0,0 20 0,9.99999999999995 30 0,10 10 0)),
	((12 12 10,11 11 10,10 10 0,12 12 10)),((12 28 10,11 11 10,12 12 10,12 28 10)),
	((9.99999999999995 30 0,11 29 10,12 28 10,9.99999999999995 30 0)),((0 20 0,2 20 10,9.99999999999995 30 0,0 20 0)),
	((10 10 0,2 20 10,0 20 0,10 10 0)),((11 11 10,2 20 10,10 10 0,11 11 10)),((12 28 10,11 29 10,11 11 10,12 28 10)),
	((9.99999999999995 30 0,2 20 10,11 29 10,9.99999999999995 30 0)),((11 11 10,11 29 10,2 20 10,11 11 10)))</screen>

	  </refsection>

	</refentry>

	<refentry id="ST_3DDifference">
	  <refnamediv>
		<refname>ST_3DDifference</refname>

		<refpurpose>Perform 3D difference</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_3DDifference</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom1</parameter></paramdef>
			<paramdef><type>geometry</type> <parameter>geom2</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>
		<para>Returns that part of geom1 that is not part of geom2.</para>
		<para role="availability" conformance="2.2.0">Availability: 2.2.0</para>
		<para>&sfcgal_required;</para>
        <para>&sqlmm_compliant; SQL-MM IEC 13249-3: 5.1</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>

	  <refsection>
	      <title>Examples</title>
	        <para>3D images were generated using PostGIS <xref linkend="ST_AsX3D" /> and rendering in HTML using <ulink url="http://www.x3dom.org">X3Dom HTML Javascript rendering library</ulink>.</para>
            <informaltable>
            <tgroup cols="2">
            <tbody>
              <row>
                <entry><para>
                <programlisting>SELECT ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(100 90)'),
 50, 'quad_segs=2'),0,0,30) AS geom1,
        ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(80 80)'),
 50, 'quad_segs=1'),0,0,30) AS geom2;
                </programlisting>
                <informalfigure>
                    <mediaobject>
                      <imageobject>
                        <imagedata fileref="images/st_3ddifference01.png" />
                      </imageobject>
                      <caption><para>Original 3D geometries overlaid. geom2 is the part that will be removed.</para></caption>
                    </mediaobject>
                  </informalfigure></para>
              </entry>
              <entry><para>
              <programlisting>SELECT ST_3DDifference(geom1,geom2)
FROM ( SELECT ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(100 90)'),
 50, 'quad_segs=2'),0,0,30) AS geom1,
        ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(80 80)'),
 50, 'quad_segs=1'),0,0,30) AS geom2 ) As t;</programlisting>
              <informalfigure>
                    <mediaobject>
                      <imageobject>
                        <imagedata fileref="images/st_3ddifference02.png" />
                      </imageobject>
                      <caption><para>What's left after removing geom2</para></caption>
                    </mediaobject>
                  </informalfigure></para>
              </entry>
              </row>
             </tbody>
            </tgroup>
            </informaltable>

	  </refsection>

      <refsection>
        <title>See Also</title>

        <para>
            <xref linkend="ST_Extrude" />,
            <xref linkend="ST_AsX3D" />,  <xref linkend="ST_3DIntersection" />
            <xref linkend="ST_3DUnion" />
        </para>
      </refsection>
	</refentry>

	<refentry id="ST_3DUnion">
	  <refnamediv>
		<refname>ST_3DUnion</refname>

		<refpurpose>Perform 3D union.</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_3DUnion</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom1</parameter></paramdef>
			<paramdef><type>geometry</type> <parameter>geom2</parameter></paramdef>
		  </funcprototype>
		  <funcprototype>
			<funcdef>geometry <function>ST_3DUnion</function></funcdef>
			<paramdef><type>geometry set</type> <parameter>g1field</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>


		<para role="availability" conformance="2.2.0">Availability: 2.2.0</para>
		<para role="availability" conformance="3.3.0">Availability: 3.3.0 aggregate variant was added</para>
		<para>&sfcgal_required;</para>
        <para>&sqlmm_compliant; SQL-MM IEC 13249-3: 5.1</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>

        <para>
            <emphasis role="bold">Aggregate variant:</emphasis>
            returns a geometry that is the 3D union of a rowset of geometries.
            The ST_3DUnion() function is an "aggregate"
            function in the terminology of PostgreSQL. That means that it
            operates on rows of data, in the same way the SUM() and AVG()
            functions do and like most aggregates, it also ignores NULL geometries.
        </para>
	  </refsection>

		  <refsection>
	      <title>Examples</title>
	        <para>3D images were generated using PostGIS <xref linkend="ST_AsX3D" /> and rendering in HTML using <ulink url="http://www.x3dom.org">X3Dom HTML Javascript rendering library</ulink>.</para>
            <informaltable>
            <tgroup cols="2">
            <tbody>
              <row>
                <entry><para>
                <programlisting>SELECT ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(100 90)'),
 50, 'quad_segs=2'),0,0,30) AS geom1,
        ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(80 80)'),
 50, 'quad_segs=1'),0,0,30) AS geom2;
                </programlisting>
                <informalfigure>
                    <mediaobject>
                      <imageobject>
                        <imagedata fileref="images/st_3ddifference01.png" />
                      </imageobject>
                      <caption><para>Original 3D geometries overlaid. geom2 is the one with transparency.</para></caption>
                    </mediaobject>
                  </informalfigure></para>
              </entry>
              <entry><para>
              <programlisting>SELECT ST_3DUnion(geom1,geom2)
FROM ( SELECT ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(100 90)'),
 50, 'quad_segs=2'),0,0,30) AS geom1,
        ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(80 80)'),
 50, 'quad_segs=1'),0,0,30) AS geom2 ) As t;</programlisting>
              <informalfigure>
                    <mediaobject>
                      <imageobject>
                        <imagedata fileref="images/st_3dunion01.png" />
                      </imageobject>
                      <caption><para>Union of geom1 and geom2</para></caption>
                    </mediaobject>
                  </informalfigure></para>
              </entry>
              </row>
             </tbody>
            </tgroup>
            </informaltable>

	  </refsection>

      <refsection>
        <title>See Also</title>

        <para>
            <xref linkend="ST_Extrude" />,
            <xref linkend="ST_AsX3D" />,  <xref linkend="ST_3DIntersection" />
            <xref linkend="ST_3DDifference" />
        </para>
      </refsection>
	</refentry>

	<refentry id="ST_AlphaShape">
	  <refnamediv>
		<refname>ST_AlphaShape</refname>

		<refpurpose>Computes an Alpha-shape enclosing a geometry</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_AlphaShape</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom</parameter></paramdef>
                        <paramdef choice="opt"><type>float </type> <parameter>alpha</parameter></paramdef>
                        <paramdef choice="opt"><type>boolean </type> <parameter>allow_holes = false</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
        <title>Description</title>

        <para>Computes the
        <ulink url="https://en.wikipedia.org/wiki/Alpha_shape">Alpha-Shape</ulink>
            of the points in a geometry.
        An alpha-shape is a (usually) concave polygonal geometry which contains all the vertices of the input,
        and whose vertices are a subset of the input vertices.
        An alpha-shape provides a closer fit to the shape of the input
        than the shape produced by the <link linkend="ST_ConvexHull">convex hull</link>.
        </para>

        <para>
        The "closeness of fit" is controlled by the <varname>alpha</varname> parameter,
        which can have values from 0 to infinity.
        Smaller alpha values produce more concave results.
        Alpha values greater than some data-dependent value produce the convex hull of the input.
        </para>

        <note><para>
        Following the CGAL implementation, the alpha value is the <emphasis>square</emphasis> of the radius of the disc used
        in the Alpha-Shape algorithm to "erode" the Delaunay Triangulation of the input points.
        See <ulink url="https://doc.cgal.org/latest/Alpha_shapes_2/index.html#Chapter_2D_Alpha_Shapes">CGAL Alpha-Shapes</ulink>
        for more information.
        This is different from the original definition of alpha-shapes,
        which defines alpha as the radius of the eroding disc.
        </para></note>

        <para>The computed shape does not contain holes unless the optional <varname>allow_holes</varname> argument is specified as true.
        </para>

        <para>
        This function effectively computes a concave hull of a geometry in a similar way to <xref linkend="ST_ConcaveHull" />,
        but uses CGAL and a different algorithm.
        </para>

		<para role="availability" conformance="3.3.0">Availability: 3.3.0 - requires SFCGAL &gt;= 1.4.1.</para>
		<para>&sfcgal_required;</para>
	  </refsection>

	  <refsection>
	      <title>Examples</title>
<para>
    <informalfigure>
      <mediaobject>
        <imageobject>
          <imagedata fileref="images/st_alphashape01.png" />
        </imageobject>
        <caption><para>Alpha-shape of a MultiPoint (same example As <xref linkend="ST_OptimalAlphaShape" />)</para></caption>
      </mediaobject>
    </informalfigure>
    <programlisting>SELECT ST_AsText(ST_AlphaShape('MULTIPOINT((63 84),(76 88),(68 73),(53 18),(91 50),(81 70),
                (88 29),(24 82),(32 51),(37 23),(27 54),(84 19),(75 87),(44 42),(77 67),(90 30),(36 61),(32 65),
                (81 47),(88 58),(68 73),(49 95),(81 60),(87 50),
                (78 16),(79 21),(30 22),(78 43),(26 85),(48 34),(35 35),(36 40),(31 79),(83 29),(27 84),(52 98),(72 95),(85 71),
                (75 84),(75 77),(81 29),(77 73),(41 42),(83 72),(23 36),(89 53),(27 57),(57 97),(27 77),(39 88),(60 81),
                (80 72),(54 32),(55 26),(62 22),(70 20),(76 27),(84 35),(87 42),(82 54),(83 64),(69 86),(60 90),(50 86),(43 80),(36 73),
                (36 68),(40 75),(24 67),(23 60),(26 44),(28 33),(40 32),(43 19),(65 16),(73 16),(38 46),(31 59),(34 86),(45 90),(64 97))'::geometry,80.2));</programlisting>
    <screen>POLYGON((89 53,91 50,87 42,90 30,88 29,84 19,78 16,73 16,65 16,53 18,43 19,
37 23,30 22,28 33,23 36,26 44,27 54,23 60,24 67,27 77,
24 82,26 85,34 86,39 88,45 90,49 95,52 98,57 97,
64 97,72 95,76 88,75 84,83 72,85 71,88 58,89 53))</screen>
        </para>
    <para>
        <informalfigure>
        <mediaobject>
            <imageobject>
            <imagedata fileref="images/st_alphashape02.png"/>
            </imageobject>
            <caption><para>Alpha-shape of a MultiPoint, allowing holes (same example as <xref linkend="ST_OptimalAlphaShape"/>)</para></caption>
        </mediaobject>
        </informalfigure>
        <programlisting>SELECT ST_AsText(ST_AlphaShape('MULTIPOINT((63 84),(76 88),(68 73),(53 18),(91 50),(81 70),(88 29),(24 82),(32 51),(37 23),(27 54),(84 19),(75 87),(44 42),(77 67),(90 30),(36 61),(32 65),(81 47),(88 58),(68 73),(49 95),(81 60),(87 50),
                (78 16),(79 21),(30 22),(78 43),(26 85),(48 34),(35 35),(36 40),(31 79),(83 29),(27 84),(52 98),(72 95),(85 71),
                (75 84),(75 77),(81 29),(77 73),(41 42),(83 72),(23 36),(89 53),(27 57),(57 97),(27 77),(39 88),(60 81),
                (80 72),(54 32),(55 26),(62 22),(70 20),(76 27),(84 35),(87 42),(82 54),(83 64),(69 86),(60 90),(50 86),(43 80),(36 73),
                (36 68),(40 75),(24 67),(23 60),(26 44),(28 33),(40 32),(43 19),(65 16),(73 16),(38 46),(31 59),(34 86),(45 90),(64 97))'::geometry, 100.1,true))</programlisting>
            <screen>POLYGON((89 53,91 50,87 42,90 30,84 19,78 16,73 16,65 16,53 18,43 19,30 22,28 33,23 36,
26 44,27 54,23 60,24 67,27 77,24 82,26 85,34 86,39 88,45 90,49 95,52 98,57 97,64 97,72 95,
76 88,75 84,83 72,85 71,88 58,89 53),(36 61,36 68,40 75,43 80,60 81,68 73,77 67,
81 60,82 54,81 47,78 43,76 27,62 22,54 32,44 42,38 46,36 61))</screen>
    </para>

    <para>
        <informalfigure>
        <mediaobject>
            <imageobject>
            <imagedata fileref="images/st_alphashape03.png"/>
            </imageobject>
            <caption><para>Alpha-shape of a MultiPoint, allowing holes (same example as <xref linkend="ST_ConcaveHull"/>)</para></caption>
        </mediaobject>
        </informalfigure>
            <programlisting>SELECT ST_AsText(ST_AlphaShape(
            'MULTIPOINT ((132 64), (114 64), (99 64), (81 64), (63 64), (57 49), (52 36), (46 20), (37 20), (26 20), (32 36), (39 55), (43 69), (50 84), (57 100), (63 118), (68 133), (74 149), (81 164), (88 180), (101 180), (112 180), (119 164), (126 149), (132 131), (139 113), (143 100), (150 84), (157 69), (163 51), (168 36), (174 20), (163 20), (150 20), (143 36), (139 49), (132 64), (99 151), (92 138), (88 124), (81 109), (74 93), (70 82), (83 82), (99 82), (112 82), (126 82), (121 96), (114 109), (110 122), (103 138), (99 151), (34 27), (43 31), (48 44), (46 58), (52 73), (63 73), (61 84), (72 71), (90 69), (101 76), (123 71), (141 62), (166 27), (150 33), (159 36), (146 44), (154 53), (152 62), (146 73), (134 76), (143 82), (141 91), (130 98), (126 104), (132 113), (128 127), (117 122), (112 133), (119 144), (108 147), (119 153), (110 171), (103 164), (92 171), (86 160), (88 142), (79 140), (72 124), (83 131), (79 118), (68 113), (63 102), (68 93), (35 45))'::geometry,102.2, true));</programlisting>
            <screen>POLYGON((26 20,32 36,35 45,39 55,43 69,50 84,57 100,63 118,68 133,74 149,81 164,88 180,
						101 180,112 180,119 164,126 149,132 131,139 113,143 100,150 84,157 69,163 51,168 36,
						174 20,163 20,150 20,143 36,139 49,132 64,114 64,99 64,90 69,81 64,63 64,57 49,52 36,46 20,37 20,26 20),
						(74 93,81 109,88 124,92 138,103 138,110 122,114 109,121 96,112 82,99 82,83 82,74 93))</screen>
    </para>
	  </refsection>
      <refsection>
		<title>See Also</title>
                <para><xref linkend="ST_ConcaveHull" />, <xref linkend="ST_OptimalAlphaShape" /></para>
	  </refsection>

	</refentry>

	<refentry id="ST_ApproximateMedialAxis">
	  <refnamediv>
		<refname>ST_ApproximateMedialAxis</refname>

		<refpurpose>Compute the approximate medial axis of an areal geometry.</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_ApproximateMedialAxis</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

    <para>
Return an approximate medial axis for the areal input based on
its straight skeleton. Uses an SFCGAL specific API when built against
a capable version (1.2.0+). Otherwise the function is just a wrapper
around ST_StraightSkeleton (slower case).
    </para>

		<para role="availability" conformance="2.2.0">Availability: 2.2.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>

	  <refsection><title>Examples</title>
			<programlisting>SELECT ST_ApproximateMedialAxis(ST_GeomFromText('POLYGON (( 190 190, 10 190, 10 10, 190 10, 190 20, 160 30, 60 30, 60 130, 190 140, 190 190 ))'));</programlisting>

		 <informaltable>
				  <tgroup cols="1">
					<tbody>
					  <row>
						<entry><para><informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_approximatemedialaxis01.png" />
							  </imageobject>
							  <caption><para>A polygon and its approximate medial axis</para></caption>
							</mediaobject>
						  </informalfigure></para>
					  </entry>
				</row>
				</tbody>
				</tgroup>
			</informaltable>
	  </refsection>

	  <refsection>
				<title>See Also</title>

				<para><xref linkend="ST_StraightSkeleton" /></para>
		</refsection>

	</refentry>

	<refentry id="ST_ConstrainedDelaunayTriangles">
		<refnamediv>
			<refname>ST_ConstrainedDelaunayTriangles</refname>

			<refpurpose>
Return a constrained Delaunay triangulation around the given input geometry.
			</refpurpose>
		</refnamediv>

		<refsynopsisdiv>
			<funcsynopsis>
			  <funcprototype>
				<funcdef>geometry <function>ST_ConstrainedDelaunayTriangles</function></funcdef>
				<paramdef><type>geometry </type> <parameter>g1</parameter></paramdef>
			  </funcprototype>
			</funcsynopsis>
		</refsynopsisdiv>

		  <refsection>
			<title>Description</title>

			<para>
Return a <ulink
url="https://en.wikipedia.org/wiki/Constrained_Delaunay_triangulation">Constrained Delaunay
triangulation</ulink> around the vertices of the input geometry.
Output is a TIN.
			</para>
			<para>&sfcgal_required;</para>
			<para role="availability" conformance="3.0.0">Availability: 3.0.0</para>
			<para>&Z_support;</para>
		</refsection>
			  <refsection>
		<title>Examples</title>
			<informaltable>
				  <tgroup cols="2">
					<tbody>
					  <row>
						  <entry><para><informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_constraineddelaunaytriangles01.png" />
							  </imageobject>
							  <caption><para>ST_ConstrainedDelaunayTriangles of 2 polygons</para></caption>
							</mediaobject>
						  </informalfigure>
						  <programlisting>
select ST_ConstrainedDelaunayTriangles(
               ST_Union(
                       'POLYGON((175 150, 20 40, 50 60, 125 100, 175 150))'::geometry,
                       ST_Buffer('POINT(110 170)'::geometry, 20)
                   )
           );
				</programlisting>
						  </para></entry>
							<entry><para><informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_constraineddelaunaytriangles02.png" />
							  </imageobject>
							  <caption><para><xref linkend="ST_DelaunayTriangles" />  of 2 polygons. Triangle edges cross polygon boundaries.</para></caption>
							</mediaobject>
						  </informalfigure>
						  <programlisting>
select ST_DelaunayTriangles(
               ST_Union(
                       'POLYGON((175 150, 20 40, 50 60, 125 100, 175 150))'::geometry,
                       ST_Buffer('POINT(110 170)'::geometry, 20)
                   )
	   );
</programlisting>
						  </para></entry>
			 </row>
		</tbody>
	</tgroup>
</informaltable>
	  </refsection>
		<refsection>
			<title>See Also</title>
			<para>
				<xref linkend="ST_DelaunayTriangles" />,
				<xref linkend="ST_TriangulatePolygon" />,
				<xref linkend="ST_Tesselate" />,
				<xref linkend="ST_ConcaveHull" />,
				<xref linkend="ST_Dump" />
			</para>
		</refsection>
	</refentry>

	<refentry id="ST_Extrude">
	  <refnamediv>
		<refname>ST_Extrude</refname>

		<refpurpose>Extrude a surface to a related volume</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_Extrude</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom</parameter></paramdef>
			<paramdef><type>float</type> <parameter>x</parameter></paramdef>
			<paramdef><type>float</type> <parameter>y</parameter></paramdef>
			<paramdef><type>float</type> <parameter>z</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

		<para role="availability" conformance="2.1.0">Availability: 2.1.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>

	  <refsection><title>Examples</title>
	        <para>3D images were generated using PostGIS <xref linkend="ST_AsX3D" /> and rendering in HTML using <ulink url="http://www.x3dom.org">X3Dom HTML Javascript rendering library</ulink>.</para>

		 <informaltable>
				  <tgroup cols="2">
					<tbody>
					  <row>
						<entry><para>
						<programlisting>SELECT ST_Buffer(ST_GeomFromText('POINT(100 90)'),
  50, 'quad_segs=2'),0,0,30);</programlisting>
						<informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_extrude01.png" />
							  </imageobject>
							  <caption><para>Original octagon formed from buffering point</para></caption>
							</mediaobject>
						  </informalfigure></para>
					  </entry>
					  <entry><para>
					  <programlisting>ST_Extrude(ST_Buffer(ST_GeomFromText('POINT(100 90)'),
 50, 'quad_segs=2'),0,0,30);</programlisting>
					  <informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_extrude02.png" />
							  </imageobject>
							  <caption><para>Hexagon extruded 30 units along Z produces a PolyhedralSurfaceZ</para></caption>
							</mediaobject>
						  </informalfigure></para>
					  </entry>
					  </row>

					<row>
						<entry><para>
						<programlisting>SELECT ST_GeomFromText('LINESTRING(50 50, 100 90, 95 150)')</programlisting>
						<informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_extrude03.png" />
							  </imageobject>
							  <caption><para>Original linestring</para></caption>
							</mediaobject>
						  </informalfigure></para>
					  </entry>
					  <entry><para>
					  <programlisting>SELECT ST_Extrude(
 ST_GeomFromText('LINESTRING(50 50, 100 90, 95 150)'),0,0,10));</programlisting>
					  <informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_extrude04.png" />
							  </imageobject>
							  <caption><para>LineString Extruded along Z produces a PolyhedralSurfaceZ</para></caption>
							</mediaobject>
						  </informalfigure></para>
					  </entry>
					  </row>
				</tbody>
				</tgroup>
			</informaltable>
	  </refsection>
        <refsection>
                <title>See Also</title>

                <para><xref linkend="ST_AsX3D" /></para>
        </refsection>
	</refentry>

	<refentry id="ST_ForceLHR">
	  <refnamediv>
		<refname>ST_ForceLHR</refname>

		<refpurpose>Force LHR orientation</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_ForceLHR</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

		<para role="availability" conformance="2.1.0">Availability: 2.1.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>

	</refentry>

    <refentry id="ST_IsPlanar">
        <refnamediv>
        <refname>ST_IsPlanar</refname>

        <refpurpose>Check if a surface is or not planar</refpurpose>
        </refnamediv>

        <refsynopsisdiv>
        <funcsynopsis>
            <funcprototype>
            <funcdef>boolean <function>ST_IsPlanar</function></funcdef>
            <paramdef><type>geometry</type> <parameter>geom</parameter></paramdef>
            </funcprototype>
        </funcsynopsis>
        </refsynopsisdiv>

        <refsection>
        <title>Description</title>

        <para role="availability" conformance="2.2.0">Availability: 2.2.0: This was documented in 2.1.0 but got accidentally left out in 2.1 release.</para>
        <para>&sfcgal_required;</para>
        <para>&Z_support;</para>
        <para>&P_support;</para>
        <para>&T_support;</para>
        </refsection>

    </refentry>

	<refentry id="ST_IsSolid">
	  <refnamediv>
		<refname>ST_IsSolid</refname>

		<refpurpose>Test if the geometry is a solid. No validity check is performed.</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>boolean <function>ST_IsSolid</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom1</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

		<para role="availability" conformance="2.2.0">Availability: 2.2.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>

	</refentry>

	<refentry id="ST_MakeSolid">
	  <refnamediv>
		<refname>ST_MakeSolid</refname>

		<refpurpose>Cast the geometry into a solid. No check is performed. To obtain a valid solid, the input geometry must be a closed Polyhedral Surface or a closed TIN.</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_MakeSolid</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom1</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

		<para role="availability" conformance="2.2.0">Availability: 2.2.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>

	</refentry>


	<refentry id="ST_MinkowskiSum">
	  <refnamediv>
		<refname>ST_MinkowskiSum</refname>

		<refpurpose>Performs Minkowski sum</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_MinkowskiSum</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom1</parameter></paramdef>
			<paramdef><type>geometry</type> <parameter>geom2</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

                <para>This function performs a 2D minkowski sum of a point, line or polygon with a polygon.</para>
                <para>A minkowski sum of two geometries A and B is the set of all points that are the sum of any point in A and B. Minkowski sums are often used in motion planning and computer-aided design. More details on <ulink url="https://en.wikipedia.org/wiki/Minkowski_addition">Wikipedia Minkowski addition</ulink>.</para>
                <para>The first parameter can be any 2D geometry (point, linestring, polygon). If a 3D geometry is passed, it will be converted to 2D by forcing Z to 0, leading to possible cases of invalidity. The second parameter must be a 2D polygon.</para>

                <para>Implementation utilizes <ulink url="http://doc.cgal.org/latest/Minkowski_sum_2/">CGAL 2D Minkowskisum</ulink>.</para>


		<para role="availability" conformance="2.1.0">Availability: 2.1.0</para>
		<para>&sfcgal_required;</para>
	  </refsection>

	 <refsection><title>Examples</title>
	 <para>Minkowski Sum of Linestring and circle polygon where Linestring cuts thru the circle</para>
            <informaltable>
                <tgroup cols="2">
                    <tbody>
                      <row>
                        <entry>
                         <para>
                            <informalfigure>
                            <mediaobject>
                              <imageobject>
                                <imagedata fileref="images/st_minkowskisum01.png" />
                              </imageobject>
                              <caption><para>Before Summing</para></caption>
                            </mediaobject>
                            </informalfigure>
                                               </para>
                        </entry>
                        <entry>
                           <para>
                            <informalfigure>
                            <mediaobject>
                              <imageobject>
                                  <imagedata fileref="images/st_minkowskisum02.png" />
                              </imageobject>
                              <caption><para>After summing</para></caption>
                            </mediaobject>
                            </informalfigure>
                                              </para>
                        </entry>
                    </row>
                    </tbody>
                </tgroup>
            </informaltable>
            <programlisting>
SELECT ST_MinkowskiSum(line, circle))
FROM (SELECT
    ST_MakeLine(ST_Point(10, 10),ST_Point(100, 100)) As line,
    ST_Buffer(ST_GeomFromText('POINT(50 50)'), 30) As circle) As foo;

-- wkt --
MULTIPOLYGON(((30 59.9999999999999,30.5764415879031 54.1472903395161,32.2836140246614 48.5194970290472,35.0559116309237 43.3328930094119,38.7867965644036 38.7867965644035,43.332893009412 35.0559116309236,48.5194970290474 32.2836140246614,54.1472903395162 30.5764415879031,60.0000000000001 30,65.8527096604839 30.5764415879031,71.4805029709527 32.2836140246614,76.6671069905881 35.0559116309237,81.2132034355964 38.7867965644036,171.213203435596 128.786796564404,174.944088369076 133.332893009412,177.716385975339 138.519497029047,179.423558412097 144.147290339516,180 150,179.423558412097 155.852709660484,177.716385975339 161.480502970953,174.944088369076 166.667106990588,171.213203435596 171.213203435596,166.667106990588 174.944088369076,
161.480502970953 177.716385975339,155.852709660484 179.423558412097,150 180,144.147290339516 179.423558412097,138.519497029047 177.716385975339,133.332893009412 174.944088369076,128.786796564403 171.213203435596,38.7867965644035 81.2132034355963,35.0559116309236 76.667106990588,32.2836140246614 71.4805029709526,30.5764415879031 65.8527096604838,30 59.9999999999999)))
            </programlisting>

        <para>Minkowski Sum of a polygon and multipoint</para>
            <informaltable>
                <tgroup cols="2">
                    <tbody>
                      <row>
                        <entry>
                         <para>
                            <informalfigure>
                            <mediaobject>
                              <imageobject>
                                <imagedata fileref="images/st_minkowskisum03.png" />
                              </imageobject>
                              <caption><para>Before Summing</para></caption>
                            </mediaobject>
                            </informalfigure>
                          </para>
                        </entry>
                        <entry>
                           <para>
                            <informalfigure>
                            <mediaobject>
                              <imageobject>
                                  <imagedata fileref="images/st_minkowskisum04.png" />
                              </imageobject>
                              <caption><para>After summing: polygon is duplicated and translated to position of points</para></caption>
                            </mediaobject>
                            </informalfigure>
                           </para>
                        </entry>
                    </row>
                    </tbody>
                </tgroup>
            </informaltable>
            <programlisting>SELECT ST_MinkowskiSum(mp, poly)
FROM (SELECT 'MULTIPOINT(25 50,70 25)'::geometry As mp,
   'POLYGON((130 150, 20 40, 50 60, 125 100, 130 150))'::geometry As poly
    ) As foo


-- wkt --
MULTIPOLYGON(
    ((70 115,100 135,175 175,225 225,70 115)),
    ((120 65,150 85,225 125,275 175,120 65))
    )
            </programlisting>
       </refsection>

	</refentry>

	<refentry id="ST_OptimalAlphaShape">
	  <refnamediv>
		<refname>ST_OptimalAlphaShape</refname>

		<refpurpose>Computes an Alpha-shape enclosing a geometry using an "optimal" alpha value.</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_OptimalAlphaShape</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom</parameter></paramdef>
                        <paramdef choice="opt"><type>boolean </type> <parameter>allow_holes = false</parameter></paramdef>
                        <paramdef choice="opt"><type>integer </type> <parameter>nb_components = 1</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
         <title>Description</title>

         <para>Computes the "optimal" alpha-shape of the points in a geometry.
            The alpha-shape is computed using a value of α chosen so that:
            </para>
        <orderedlist>
            <listitem>
                <para>the number of polygon elements is equal to or smaller than <varname>nb_components</varname>
                    (which defaults to 1)</para>
            </listitem>
            <listitem>
                <para>all input points are contained in the shape</para>
            </listitem>
        </orderedlist>
        <para>
            The result will not contain holes unless the optional <varname>allow_holes</varname> argument is specified as true.
            </para>

		<para role="availability" conformance="3.3.0">Availability: 3.3.0 - requires SFCGAL &gt;= 1.4.1.</para>
		<para>&sfcgal_required;</para>
	  </refsection>

	  <refsection>
	    <title>Examples</title>
        <para>
            <informalfigure>
            <mediaobject>
                <imageobject>
                <imagedata fileref="images/st_optimalalphashape01.png" />
                </imageobject>
                <caption><para>Optimal alpha-shape of a MultiPoint (same example as <xref linkend="ST_AlphaShape" />)</para></caption>
            </mediaobject>
            </informalfigure>
                <programlisting>SELECT ST_AsText(ST_OptimalAlphaShape('MULTIPOINT((63 84),(76 88),(68 73),(53 18),(91 50),(81 70),
                (88 29),(24 82),(32 51),(37 23),(27 54),(84 19),(75 87),(44 42),(77 67),(90 30),(36 61),(32 65),
                (81 47),(88 58),(68 73),(49 95),(81 60),(87 50),
                (78 16),(79 21),(30 22),(78 43),(26 85),(48 34),(35 35),(36 40),(31 79),(83 29),(27 84),(52 98),(72 95),(85 71),
                (75 84),(75 77),(81 29),(77 73),(41 42),(83 72),(23 36),(89 53),(27 57),(57 97),(27 77),(39 88),(60 81),
                (80 72),(54 32),(55 26),(62 22),(70 20),(76 27),(84 35),(87 42),(82 54),(83 64),(69 86),(60 90),(50 86),(43 80),(36 73),
                (36 68),(40 75),(24 67),(23 60),(26 44),(28 33),(40 32),(43 19),(65 16),(73 16),(38 46),(31 59),(34 86),(45 90),(64 97))'::geometry));</programlisting>
                <screen>POLYGON((89 53,91 50,87 42,90 30,88 29,84 19,78 16,73 16,65 16,53 18,43 19,37 23,30 22,28 33,23 36,
                26 44,27 54,23 60,24 67,27 77,24 82,26 85,34 86,39 88,45 90,49 95,52 98,57 97,64 97,72 95,76 88,75 84,75 77,83 72,85 71,83 64,88 58,89 53))</screen>
        </para>

        <para>
            <informalfigure>
            <mediaobject>
                <imageobject>
                <imagedata fileref="images/st_optimalalphashape02.png" />
                </imageobject>
                <caption><para>Optimal alpha-shape of a MultiPoint, allowing holes (same example as <xref linkend="ST_AlphaShape" />)</para></caption>
            </mediaobject>
            </informalfigure>
                <programlisting>SELECT ST_AsText(ST_OptimalAlphaShape('MULTIPOINT((63 84),(76 88),(68 73),(53 18),(91 50),(81 70),(88 29),(24 82),(32 51),(37 23),(27 54),(84 19),(75 87),(44 42),(77 67),(90 30),(36 61),(32 65),(81 47),(88 58),(68 73),(49 95),(81 60),(87 50),
                (78 16),(79 21),(30 22),(78 43),(26 85),(48 34),(35 35),(36 40),(31 79),(83 29),(27 84),(52 98),(72 95),(85 71),
                (75 84),(75 77),(81 29),(77 73),(41 42),(83 72),(23 36),(89 53),(27 57),(57 97),(27 77),(39 88),(60 81),
                (80 72),(54 32),(55 26),(62 22),(70 20),(76 27),(84 35),(87 42),(82 54),(83 64),(69 86),(60 90),(50 86),(43 80),(36 73),
                (36 68),(40 75),(24 67),(23 60),(26 44),(28 33),(40 32),(43 19),(65 16),(73 16),(38 46),(31 59),(34 86),(45 90),(64 97))'::geometry, allow_holes => true));</programlisting>
                <screen>POLYGON((89 53,91 50,87 42,90 30,88 29,84 19,78 16,73 16,65 16,53 18,43 19,37 23,30 22,28 33,23 36,26 44,27 54,23 60,24 67,27 77,24 82,26 85,34 86,39 88,45 90,49 95,52 98,57 97,64 97,72 95,76 88,75 84,75 77,83 72,85 71,83 64,88 58,89 53),(36 61,36 68,40 75,43 80,50 86,60 81,68 73,77 67,81 60,82 54,81 47,78 43,81 29,76 27,70 20,62 22,55 26,54 32,48 34,44 42,38 46,36 61))</screen>
        </para>

	  </refsection>
      <refsection>
		<title>See Also</title>
		<para><xref linkend="ST_ConcaveHull" />, <xref linkend="ST_AlphaShape" /></para>
	  </refsection>

	</refentry>

	<refentry id="ST_Orientation">
	  <refnamediv>
		<refname>ST_Orientation</refname>

		<refpurpose>Determine surface orientation</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>integer <function>ST_Orientation</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

		<para>The function only applies to polygons. It returns -1 if the polygon is counterclockwise oriented and 1 if the polygon is clockwise oriented.</para>

		<para role="availability" conformance="2.1.0">Availability: 2.1.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
	  </refsection>

	</refentry>

    <refentry id="ST_StraightSkeleton">
	  <refnamediv>
		<refname>ST_StraightSkeleton</refname>

		<refpurpose>Compute a straight skeleton from a geometry</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_StraightSkeleton</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

		<para role="availability" conformance="2.1.0">Availability: 2.1.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>

	  <refsection><title>Examples</title>
			<programlisting>SELECT ST_StraightSkeleton(ST_GeomFromText('POLYGON (( 190 190, 10 190, 10 10, 190 10, 190 20, 160 30, 60 30, 60 130, 190 140, 190 190 ))'));</programlisting>

		 <informaltable>
				  <tgroup cols="2">
					<tbody>
					  <row>
						<entry><para><informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_straightskeleton01.png" />
							  </imageobject>
							  <caption><para>Original polygon</para></caption>
							</mediaobject>
						  </informalfigure></para>
					  </entry>
					  <entry><para><informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_straightskeleton02.png" />
							  </imageobject>
							  <caption><para>Straight Skeleton of polygon</para></caption>
							</mediaobject>
						  </informalfigure></para>
					  </entry>
				</row>
				</tbody>
				</tgroup>
			</informaltable>
	  </refsection>

	</refentry>


	<refentry id="ST_Tesselate">
	  <refnamediv>
		<refname>ST_Tesselate</refname>

		<refpurpose>Perform surface Tesselation of a polygon or polyhedralsurface and returns as a TIN or collection of TINS</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>geometry <function>ST_Tesselate</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>
		<para>Takes as input a surface such a  MULTI(POLYGON) or POLYHEDRALSURFACE and returns a TIN representation via the process of tessellation using triangles.</para>
		<para role="availability" conformance="2.1.0">Availability: 2.1.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
	  </refsection>

	  <refsection><title>Examples</title>
	  <informaltable>
				  <tgroup cols="2">
					<tbody>
					  <row>
						<entry><para>
						<programlisting>SELECT ST_GeomFromText('POLYHEDRALSURFACE Z( ((0 0 0, 0 0 1, 0 1 1, 0 1 0, 0 0 0)),
		((0 0 0, 0 1 0, 1 1 0, 1 0 0, 0 0 0)), ((0 0 0, 1 0 0, 1 0 1, 0 0 1, 0 0 0)),
		((1 1 0, 1 1 1, 1 0 1, 1 0 0, 1 1 0)),
		((0 1 0, 0 1 1, 1 1 1, 1 1 0, 0 1 0)), ((0 0 1, 1 0 1, 1 1 1, 0 1 1, 0 0 1)) )');</programlisting>
						<informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_tesselate01.png" />
							  </imageobject>
							  <caption><para>Original Cube</para></caption>
							</mediaobject>
						  </informalfigure></para>
					  </entry>
					  <entry><para>
					  <programlisting>SELECT ST_Tesselate(ST_GeomFromText('POLYHEDRALSURFACE Z( ((0 0 0, 0 0 1, 0 1 1, 0 1 0, 0 0 0)),
	((0 0 0, 0 1 0, 1 1 0, 1 0 0, 0 0 0)), ((0 0 0, 1 0 0, 1 0 1, 0 0 1, 0 0 0)),
	((1 1 0, 1 1 1, 1 0 1, 1 0 0, 1 1 0)),
	((0 1 0, 0 1 1, 1 1 1, 1 1 0, 0 1 0)), ((0 0 1, 1 0 1, 1 1 1, 0 1 1, 0 0 1)) )'));</programlisting></para>
		<para>ST_AsText output:</para>
		<screen>TIN Z (((0 0 0,0 0 1,0 1 1,0 0 0)),((0 1 0,0 0 0,0 1 1,0 1 0)),
	((0 0 0,0 1 0,1 1 0,0 0 0)),
	((1 0 0,0 0 0,1 1 0,1 0 0)),((0 0 1,1 0 0,1 0 1,0 0 1)),
	((0 0 1,0 0 0,1 0 0,0 0 1)),
	((1 1 0,1 1 1,1 0 1,1 1 0)),((1 0 0,1 1 0,1 0 1,1 0 0)),
	((0 1 0,0 1 1,1 1 1,0 1 0)),((1 1 0,0 1 0,1 1 1,1 1 0)),
	((0 1 1,1 0 1,1 1 1,0 1 1)),((0 1 1,0 0 1,1 0 1,0 1 1)))</screen>
		       <para> <informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_tesselate02.png" />
							  </imageobject>
							  <caption><para>Tesselated Cube with triangles colored</para></caption>
							</mediaobject>
						  </informalfigure></para>
					  </entry>
					  </row>

					  <row>
						<entry><para>
						<programlisting>SELECT 'POLYGON (( 10 190, 10 70, 80 70, 80 130, 50 160, 120 160, 120 190, 10 190 ))'::geometry;</programlisting>
						<informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_tesselate03.png" />
							  </imageobject>
							  <caption><para>Original polygon</para></caption>
							</mediaobject>
						  </informalfigure></para>
					  </entry>
					  <entry><para>
					  <programlisting>SELECT
	ST_Tesselate('POLYGON (( 10 190, 10 70, 80 70, 80 130, 50 160, 120 160, 120 190, 10 190 ))'::geometry);</programlisting>
	</para>
	<para>ST_AsText output</para>
	<screen>TIN(((80 130,50 160,80 70,80 130)),((50 160,10 190,10 70,50 160)),
	 ((80 70,50 160,10 70,80 70)),((120 160,120 190,50 160,120 160)),
 ((120 190,10 190,50 160,120 190)))</screen>
					  <para><informalfigure>
							<mediaobject>
							  <imageobject>
								<imagedata fileref="images/st_tesselate04.png" />
							  </imageobject>
							  <caption><para>Tesselated Polygon</para></caption>
							</mediaobject>
						  </informalfigure></para>
					  </entry>
					  </row>
				</tbody>
				</tgroup>
			</informaltable>
		</refsection>

		<refsection>
                <title>See Also</title>

                <para><xref linkend="ST_ConstrainedDelaunayTriangles" />, <xref linkend="ST_DelaunayTriangles" /></para>
          </refsection>


	</refentry>

	<refentry id="ST_Volume">
	  <refnamediv>
		<refname>ST_Volume</refname>

		<refpurpose>Computes the volume of a 3D solid. If applied to surface (even closed) geometries will return 0.</refpurpose>
	  </refnamediv>

	  <refsynopsisdiv>
		<funcsynopsis>
		  <funcprototype>
			<funcdef>float <function>ST_Volume</function></funcdef>
			<paramdef><type>geometry</type> <parameter>geom1</parameter></paramdef>
		  </funcprototype>
		</funcsynopsis>
	  </refsynopsisdiv>

	  <refsection>
		<title>Description</title>

		<para role="availability" conformance="2.2.0">Availability: 2.2.0</para>
		<para>&sfcgal_required;</para>
		<para>&Z_support;</para>
		<para>&P_support;</para>
		<para>&T_support;</para>
        <para>&sqlmm_compliant; SQL-MM IEC 13249-3: 9.1 (same as ST_3DVolume)</para>
	  </refsection>

	  <refsection>
	      <title>Example</title>
	      <para>When closed surfaces are created with WKT, they are treated as areal rather than solid.  To make them solid, you need to use <xref linkend="ST_MakeSolid" />.
	      Areal geometries have no volume.  Here is an example to demonstrate.</para>
	      <programlisting>SELECT ST_Volume(geom) As cube_surface_vol,
	ST_Volume(ST_MakeSolid(geom)) As solid_surface_vol
  FROM (SELECT 'POLYHEDRALSURFACE( ((0 0 0, 0 0 1, 0 1 1, 0 1 0, 0 0 0)),
    ((0 0 0, 0 1 0, 1 1 0, 1 0 0, 0 0 0)),
    ((0 0 0, 1 0 0, 1 0 1, 0 0 1, 0 0 0)),
    ((1 1 0, 1 1 1, 1 0 1, 1 0 0, 1 1 0)),
    ((0 1 0, 0 1 1, 1 1 1, 1 1 0, 0 1 0)),
    ((0 0 1, 1 0 1, 1 1 1, 0 1 1, 0 0 1)) )'::geometry) As f(geom);

 cube_surface_vol | solid_surface_vol
------------------+-------------------
                0 |                 1
	      </programlisting>
	  </refsection>

	  <refsection>
		<title>See Also</title>

		<para><xref linkend="ST_3DArea" />, <xref linkend="ST_MakeSolid" />, <xref linkend="ST_IsSolid" /></para>
	  </refsection>

	</refentry>


</sect1>
